DataStructures-Algorithms-ML-concepts-using-Python

Here, you will find the explanation of Data Structures and Algorithms for Machine Learning Models using Python.

What is Gradient Descent?

• Iterative Algorithm (An iterative algorithm executes steps in iterations. It aims to find successive approximations in sequence to reach a solution. They are most commonly used in linear programs where large numbers of variables are involved.) https://web.stanford.edu/group/sisl/k12/optimization/MO-unit1-pdfs/1.8iterativeloops.pdf

Objective : Gradient descent algorithm is an iterative process that takes us to the minimum of a function.

Mathematically,

• used to find a minimum of a differentiable function. https://en.wikipedia.org/wiki/Differentiable_function

Why do we need this in ML?

• to minimize the "Cost Function" and to find the corresponding "OPTIMAL PARAMETERS' (Optimization)!

What is a Cost Function then?

• A function to measure the deviation of our Model's Prediction from the ground truth!

Which ML models implement Gradient Descent?

• Linear Regression Models (Simple, Multiple)
• Logistic Regression Models

Types of Gradient Descent in ML?

• Batch Gradient Descent
• Stochastic Gradient Descent

Gradient Descent Procedure:

The procedure starts off with initial values for the coefficient or coefficients for the function. These could be 0.0 or a small random value.

coefficient = 0.0

The cost of the coefficients is evaluated by plugging them into the function and calculating the cost.

cost = f(coefficient)

or

cost = evaluate(f(coefficient))

The derivative of the cost is calculated. The derivative is a concept from calculus and refers to the slope of the function at a given point. We need to know the slope so that we know the direction (sign) to move the coefficient values in order to get a lower cost on the next iteration.

delta = derivative(cost)

Now that we know from the derivative which direction is downhill, we can now update the coefficient values. A learning rate parameter (alpha) must be specified that controls how much the coefficients can change on each update.

coefficient = coefficient – (alpha * delta)

This process is repeated until the cost of the coefficients (cost) is 0.0 or close enough to zero to be good enough.

You can see how simple gradient descent is. It does require you to know the gradient of your cost function or the function you are optimizing, but besides that, it’s very straightforward. Next we will see how we can use this in machine learning algorithms.

Batch Gradient Descent for Machine Learning:

The goal of all supervised machine learning algorithms is to best estimate a target function (f) that maps input data (X) onto output variables (Y). This describes all classification and regression problems.

Some machine learning algorithms have coefficients that characterize the algorithm's estimate for the target function (f). Different algorithms have different representations and different coefficients, but many of them require a process of optimization to find the set of coefficients that result in the best estimate of the target function.

Common examples of algorithms with coefficients that can be optimized using gradient descent are Linear Regression and Logistic Regression.

The evaluation of how close a fit a machine learning model is to estimates the target function which can be calculated a number of different ways, often specific to the machine learning algorithm. The cost function involves evaluating the coefficients in the machine learning model by calculating a prediction for the model for each training instance in the dataset and comparing the predictions to the actual output values and calculating a sum or average error (such as the Sum of Squared Residuals or SSR in the case of linear regression).

From the cost function, a derivative can be calculated for each coefficient so that it can be updated using exactly the update equation described above.

The cost is calculated for a machine learning algorithm over the entire training dataset for each iteration of the gradient descent algorithm. One iteration of the algorithm is called one batch and this form of gradient descent is referred to as batch gradient descent.

Batch gradient descent is the most common form of gradient descent described in machine learning.

Stochastic Gradient Descent for Machine Learning :

Gradient descent can be slow to run on very large datasets.

Because one iteration of the gradient descent algorithm requires a prediction for each instance in the training dataset, it can take a long time when you have many millions of instances.

In situations when you have large amounts of data, you can use a variation of gradient descent called stochastic gradient descent.

In this variation, the gradient descent procedure described above is run but the update to the coefficients is performed for each training instance, rather than at the end of the batch of instances.

The first step of the procedure requires that the order of the training dataset is randomized. This is to mix up the order that updates are made to the coefficients. Because the coefficients are updated after every training instance, the updates will be noisy jumping all over the place, and so will the corresponding cost function. By mixing up the order for the updates to the coefficients, it harnesses this random walk and avoids it getting distracted or stuck.

The update procedure for the coefficients is the same as that above, except the cost is not summed over all training patterns, but instead calculated for one training pattern.

The learning can be much faster with stochastic gradient descent for very large training datasets and often you only need a small number of passes through the dataset to reach a good or good enough set of coefficients, e.g. 1-to-10 passes through the dataset.

I have explained it with an example here:

Stochastic Gradient Descent From Scratch

This notebook illustrates the nature of the Stochastic Gradient Descent (SGD) and walks through all the necessary steps to create SGD from scratch in Python. Gradient Descent is an essential part of many machine learning algorithms, including neural networks. To understand how it works you will need some basic math and logical thinking. Though a stronger math background would be preferable to understand derivatives, I will try to explain them as simple as possible.

We will work with the California housing dataset and perform a linear regression to predict apartment prices based on the median income in the block. We will start from the simple linear regression and gradually finish with Stochastic Gradient Descent. So let's get started.

Importing Libraries

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sklearn.datasets import fetch_california_housing
from sklearn.metrics import mean_squared_error

California Housing Dataset

Scikit-learn comes with wide variety of datasets for regression, classification and other problems. Lets load our data into pandas dataframe and take a look.

housing_data = fetch_california_housing()

Features = pd.DataFrame(housing_data.data, columns=housing_data.feature_names)
Target = pd.DataFrame(housing_data.target, columns=['Target'])

df = Features.join(Target)

Features as MedInc and Target were scaled to some degree.

df.corr()

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	MedInc	HouseAge	AveRooms	AveBedrms	Population	AveOccup	Latitude	Longitude	Target
MedInc	1.000000	-0.119034	0.326895	-0.062040	0.004834	0.018766	-0.079809	-0.015176	0.688075
HouseAge	-0.119034	1.000000	-0.153277	-0.077747	-0.296244	0.013191	0.011173	-0.108197	0.105623
AveRooms	0.326895	-0.153277	1.000000	0.847621	-0.072213	-0.004852	0.106389	-0.027540	0.151948
AveBedrms	-0.062040	-0.077747	0.847621	1.000000	-0.066197	-0.006181	0.069721	0.013344	-0.046701
Population	0.004834	-0.296244	-0.072213	-0.066197	1.000000	0.069863	-0.108785	0.099773	-0.024650
AveOccup	0.018766	0.013191	-0.004852	-0.006181	0.069863	1.000000	0.002366	0.002476	-0.023737
Latitude	-0.079809	0.011173	0.106389	0.069721	-0.108785	0.002366	1.000000	-0.924664	-0.144160
Longitude	-0.015176	-0.108197	-0.027540	0.013344	0.099773	0.002476	-0.924664	1.000000	-0.045967
Target	0.688075	0.105623	0.151948	-0.046701	-0.024650	-0.023737	-0.144160	-0.045967	1.000000

Preprocessing: Removing Outliers and Scaling

df[['MedInc', 'Target']].describe()[1:] #.style.highlight_max(axis=0)

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	MedInc	Target
mean	3.482030	1.722805
std	1.364922	0.749957
min	0.499900	0.149990
25%	2.452025	1.119000
50%	3.303600	1.635000
75%	4.346050	2.256000
max	7.988700	3.499000

It seems that Target has some outliers (as well as MedInc), because 75% of the data has price less than 2.65, but maximum price go as high as 5. We're going to remove extremely expensive houses as they will add unnecessary noize to the data.

df = df[df.Target < 3.5]
df = df[df.MedInc < 8]

Removed Outliers

df[['MedInc', 'Target']].describe()[1:]

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	MedInc	Target
mean	3.482030	1.722805
std	1.364922	0.749957
min	0.499900	0.149990
25%	2.452025	1.119000
50%	3.303600	1.635000
75%	4.346050	2.256000
max	7.988700	3.499000

We will also scale MedInc and Target variables to [0-1].

def scale(x):
    min = x.min()
    max = x.max()
    return pd.Series([(i - min)/(max - min) for i in x])

X = scale(df.MedInc)
y = scale(df.Target)

X.max(), y.max() # features are scaled now

(1.0, 1.0)

Correlation Between Price and Income

Visually we can determine what kind of accuracy we can expect from the models.

plt.figure(figsize=(16,6))
plt.rcParams['figure.dpi'] = 227
plt.style.use('seaborn-whitegrid')
plt.scatter(X, y, label='Data', c='#388fd8', s=6)
plt.title('Positive Correlation Between Income and House Price', fontSize=15)
plt.xlabel('Income', fontSize=12)
plt.ylabel('House Price', fontSize=12)
plt.legend(frameon=True, loc=1, fontsize=10, borderpad=.6)
plt.tick_params(direction='out', length=6, color='#a0a0a0', width=1, grid_alpha=.6)
plt.show()

Data is quite sparse, but we can still observe some linearity.

Simple Linear Regression

Simple linear regression can be described by only two parameters: slope m and intercept b, where x is our median income. Lets take a look at the formulas below:

$$\hat{y} = mx + b$$

$$m = \frac{\overline{x}\overline{y}-\overline{xy}}{(\overline{x})^2 - \overline{x^2}} \quad \textrm{and} \quad b = y-mx$$

If we want to add some other features, like size of the apartment, our formula would look like this: $\hat{y} = m_1x_1 + m_2x_2 + b$, where $m_1$ and $m_2$ are slopes for each feature $x_1$ and $x_2$. In this case we would call it multiple linear regression, but we could no longer use formulas above.

class SimpleLinearRegression:
        
    def fit(self, X, y):
        self.X = X
        self.y = y
        self.m = ((np.mean(X) * np.mean(y) - np.mean(X*y)) / ((np.mean(X)**2) - np.mean(X**2)))
        self.b = np.mean(y) - self.m * np.mean(X)
    
    def coeffs(self):
        return self.m, self.b
    
    def predict(self):
        self.y_pred = self.m * self.X + self.b
        return self.y_pred
    
    def r_squared(self):
        self.y_mean = np.full((len(self.y)), mean(self.y))
        err_reg = sum((self.y - self.y_pred)**2)
        err_y_mean = sum((self.y - self.y_mean)**2)
        return (1 - (err_reg/err_y_mean))

def plot_regression(X, y, y_pred, log=None, title="Linear Regression"):
    
    plt.figure(figsize=(16,6))
    plt.rcParams['figure.dpi'] = 227
    plt.scatter(X, y, label='Data', c='#388fd8', s=6)
    if log != None:
        for i in range(len(log)):
            plt.plot(X, log[i][0]*X + log[i][1], lw=1, c='#caa727', alpha=0.35)
    plt.plot(X, y_pred, c='#ff7702', lw=3, label='Regression')
    plt.title(title, fontSize=14)
    plt.xlabel('Income', fontSize=11)
    plt.ylabel('Price', fontSize=11)
    plt.legend(frameon=True, loc=1, fontsize=10, borderpad=.6)
    plt.tick_params(direction='out', length=6, color='#a0a0a0', width=1, grid_alpha=.6)
    plt.show()

X = df.MedInc
y = df.Target

lr = SimpleLinearRegression()

lr.fit(X, y)

y_pred = lr.predict()

print("MSE:",mean_squared_error(y, y_pred))
plot_regression(X, y, y_pred, title="Linear Regression")

MSE: 0.34320521502255963

Result of our model is the regression line. Just by looking at the graph we can tell that data points go well above and beyond our line, making predictions approximate.

Multiple Linear Regression with Least Squares

Similar to from sklearn.linear_model import LinearRegression, we can calculate coefficients with Least Squares method. Numpy can calculate this formula almost instantly (of course depends on the amount of data) and precise.

$$ m =(A^TA)^{-1} A^Ty $$

$$m - parameters, : A - data, : y - target$$

X = df.drop('Target', axis=1) # matrix A, or all the features
y = df.Target

class MultipleLinearRegression:
    '''
    Multiple Linear Regression with Least Squares    
    '''    
    def fit(self, X, y):
        X = np.array(X)
        y = np.array(y)
        self.coeffs = np.linalg.inv(X.T.dot(X)).dot(X.T).dot(y)
        
    def predict(self, X):
        X = np.array(X)
        result = np.zeros(len(X))
        for i in range(X.shape[1]):
            result += X[:, i] * self.coeffs[i]
        return result
    
    def coeffs(self):
        return self.coeffs

mlp = MultipleLinearRegression()

mlp.fit(X, y)

y_pred = mlp.predict(X)

mean_squared_error(y, y_pred)

0.2912984534321039

Gradient Descent

Abstract

The idea behind gradient descent is simple - by gradually tuning parameters, such as slope (m) and the intercept (b) in our regression function y = mx + b, we minimize cost. By cost, we usually mean some kind of a function that tells us how far off our model predicted result. For regression problems we often use mean squared error (MSE) cost function. If we use gradient descent for the classification problem, we will have a different set of parameters to tune.

$$ MSE = \frac{1}{n}\sum_{i=1}^{n} (y_i - \hat{y_i})^2 \quad \textrm{where} \quad \hat{y_i} = mx_i + b $$

Now we have to figure out how to tweak parameters m and b to reduce MSE.

Partial Derivatives

We use partial derivatives to find how each individual parameter affects MSE, so that's where word partial comes from. In simple words, we take the derivative with respect to m and b separately. Take a look at the formula below. It looks almost exactly the same as MSE, but this time we added f(m, b) to it. It essentially changes nothing, except now we can plug m and b numbers into it and calculate the result.

$$𝑓(𝑚,𝑏)= \frac{1}{n}\sum_{i=1}^{n}(y_i - (mx_i+b))^2$$

This formula (or better say function) is better representation for further calculations of partial derivatives. We can ignore sum for now and what comes before that and focus only on $y - (mx + b)^2$.

Partical Derivative With Respect to m

With respect to m means we derive parameter m and basically ignore what is going on with b, or we can say its 0. To derive with respect to m we will use chain rule.

$$ [f(g(x))]' = f'(g(x)) * g(x)' : - \textrm{chain rule}$$

Chain rule applies when one function sits inside of another. If you're new to this, you'd be surprised that $()^2$ is outside function, and $y-(\boldsymbol{m}x+b)$ sits inside it. So, the chain rule says that we should take a derivative of outside function, keep inside function unchanged and then multiply by derivative of the inside function. Lets write these steps down:

$$ (y - (mx + b))^2 $$

1. Derivative of $()^2$ is $2()$, same as $x^2$ becomes $2x$
2. We do nothing with $y - (mx + b)$, so it stays the same
3. Derivative of $y - (mx + b)$ with respect to m is $(0 - (x + 0))$ or $-x$, because y and b are constants, they become 0, and derivative of mx is x

Multiply all parts we get following: $2 * (y - (mx+b)) * -x$. Looks nicer if we move -x to the left: $-2x *(y-(mx+b))$. There we have it. The final version of our derivative is the following:

$$\frac{\partial f}{\partial m} = \frac{1}{n}\sum_{i=1}^{n}-2x_i(y_i - (mx_i+b))$$

Here, $\frac{df}{dm}$ means we find partial derivative of function f (we mentioned it earlier) with respect to m. We plug our derivative to the summation and we're done.

Partical Derivative With Respect to b

Same rules apply to the derivative with respect to b.

1. $()^2$ becomes $2()$, same as $x^2$ becomes $2x$
2. $y - (mx + b)$ stays the same
3. $y - (mx + b)$ becomes $(0 - (0 + 1))$ or $-1$, because y and mx are constants, they become 0, and derivative of b is 1

Multiply all the parts together and we get $-2(y-(mx+b))$

$$\frac{\partial f}{\partial b} = \frac{1}{n}\sum_{i=1}^{n}-2(y_i - (mx_i+b))$$

Final Function

Few details we should discuss befor jumping into code:

1. Gradient descent is an iterative process and with each iteration (epoch) we slightly minimizing MSE, so each time we use our derived functions to update parameters m and b
2. Because its iterative, we should choose how many iterations we take, or make algorithm stop when we approach minima of MSE. In other words when algorithm is no longer improving MSE, we know it reached minimum.
3. Gradient descent has an additional parameter learning rate (lr), which helps control how fast or slow algorithm going towards minima of MSE

Thats about it. So you can already understand that Gradient Descent for the most part is just process of taking derivatives and using them over and over to minimize function.

def gradient_descent(X, y, lr=0.05, epoch=10):
    
    '''
    Gradient Descent for a single feature
    '''
    
    m, b = 0.2, 0.2 # parameters
    log, mse = [], [] # lists to store learning process
    N = len(X) # number of samples
    
    for _ in range(epoch):
                
        f = y - (m*X + b)
    
        # Updating m and b
        m -= lr * (-2 * X.dot(f).sum() / N)
        b -= lr * (-2 * f.sum() / N)
        
        log.append((m, b))
        mse.append(mean_squared_error(y, (m*X + b)))        
    
    return m, b, log, mse

Predicting House Price With Gradient Descent

X = df.MedInc
y = df.Target

m, b, log, mse = gradient_descent(X, y, lr=0.01, epoch=100)

y_pred = m*X + b

print("MSE:",mean_squared_error(y, y_pred))
plot_regression(X, y, y_pred, log=log, title="Linear Regression with Gradient Descent")

plt.figure(figsize=(16,3))
plt.rcParams['figure.dpi'] = 227
plt.plot(range(len(mse)), mse)
plt.title('Gradient Descent Optimization', fontSize=14)
plt.xlabel('Epochs')
plt.ylabel('MSE')
plt.show()

MSE: 0.3493097403876614

Stochastic Gradient Descent

Stochastic Gradient Descent works almost the same as Gradient Descent (also called Batch Gradient Descent), but instead of training on entire dataset, it picks only one sample to update m and b parameters, which makes it much faster. In the function below I made possible to change sample size (batch_size), because sometimes its better to use more than one sample at a time.

def SGD(X, y, lr=0.05, epoch=10, batch_size=1):
        
    '''
    Stochastic Gradient Descent for a single feature
    '''
    
    m, b = 0.5, 0.5 # initial parameters
    log, mse = [], [] # lists to store learning process
    
    for _ in range(epoch):
        
        indexes = np.random.randint(0, len(X), batch_size) # random sample
        
        Xs = np.take(X, indexes)
        ys = np.take(y, indexes)
        N = len(Xs)
        
        f = ys - (m*Xs + b)
        
        # Updating parameters m and b
        m -= lr * (-2 * Xs.dot(f).sum() / N)
        b -= lr * (-2 * f.sum() / N)
        
        log.append((m, b))
        mse.append(mean_squared_error(y, m*X+b))        
    
    return m, b, log, mse

m, b, log, mse = SGD(X, y, lr=0.01, epoch=100, batch_size=2)

y_pred = m*X + b

print("MSE:",mean_squared_error(y, y_pred))
plot_regression(X, y, y_pred, log=log, title="Linear Regression with SGD")

plt.figure(figsize=(16,3))
plt.rcParams['figure.dpi'] = 227
plt.plot(range(len(mse)), mse)
plt.title('SGD Optimization', fontSize=14)
plt.xlabel('Epochs', fontSize=11)
plt.ylabel('MSE', fontSize=11)
plt.show()

MSE: 0.3462919845446769

We can observe how regression line went up and down to find right parameters and MSE not as smooth as regular gradient descent.

Speed Test for Gradient Descent vs SGD

X = df.MedInc
y = df.Target

X = np.concatenate((X,X,X,X,X,X,X,X,X,X,X,X,X,X,X,X,X))
y = np.concatenate((y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y))

X.shape, y.shape

((304946,), (304946,))

%timeit SGD(X, y, lr=0.01, epoch=1000, batch_size=1)

1.22 s ± 8.95 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)

%timeit gradient_descent(X, y, lr=0.01, epoch=1000)

2.02 s ± 79.4 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)

Conclusion

1. SGD is twice as fast as Gradient Descent (also called Batch Gradient Descent)
2. On sparse data, we can increase the batch size to speed up learning process. It's not a pure form of SGD, but we can call it a mini-batch SGD
3. Smaller learning rate helps to prevent overfitting but can be adjusted accordingly

Tips for Gradient Descent :

This section lists some tips and tricks for getting the most out of the gradient descent algorithm for machine learning.

Plot Cost versus Time: Collect and plot the cost values calculated by the algorithm each iteration. The expectation for a well performing gradient descent run is a decrease in cost each iteration. If it does not decrease, try reducing your learning rate. Learning Rate: The learning rate value is a small real value such as 0.1, 0.001 or 0.0001. Try different values for your problem and see which works best. Rescale Inputs: The algorithm will reach the minimum cost faster if the shape of the cost function is not skewed and distorted. You can achieve this by rescaling all of the input variables (X) to the same range, such as [0, 1] or [-1, 1]. Few Passes: Stochastic gradient descent often does not need more than 1-to-10 passes through the training dataset to converge on good or good enough coefficients. Plot Mean Cost: The updates for each training dataset instance can result in a noisy plot of cost over time when using stochastic gradient descent. Taking the average over 10, 100, or 1000 updates can give you a better idea of the learning trend for the algorithm.

Summary:

In this file you discovered gradient descent for machine learning. You learned that:

Optimization is a big part of machine learning. Gradient descent is a simple optimization procedure that you can use with many machine learning algorithms. Batch gradient descent refers to calculating the derivative from all training data before calculating an update. Stochastic gradient descent refers to calculating the derivative from each training data instance and calculating the update immediately.

Example of a Real-world application: Agile (SDLC) is a pretty well-known term in the software development process. The basic idea behind it is simple: build something quickly, ➡️ get it out there, ➡️ get some feedback ➡️ make changes depending upon the feedback ➡️ repeat the process. The goal is to get the product near the user and guide you with feedback to obtain the best possible product with the least error. Also, the steps taken for improvement need to be small and should constantly involve the user. In a way, an Agile software development process involves rapid iterations. The idea of — start with a solution as soon as possible, measure and iterate as frequently as possible, is Gradient descent under the hood.